MAHLER MEASURE AND INTEGRALS OF HYPERGEOMETRIC FUNCTIONS

Boyd studied in [4] the Mahler measure of some families of elliptic curves given by reciprocal polynomials depending on a real parameter l. These curves are of genus 1 except for the singular values of the parameter l and do not vanish on the torus for l large enough. Using the Mahler measure, we prove that the Picard-Fuchs equation associated to one Boyd's family admits explicit solutions at the singularities of the equation and that we can express certain Dirichlet L-series L(chi, 2) as integrals of functions related to the hypergeometric functions F(1/2, 1/2, 1; z).